SKM 2021 – scientific programme

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DY: Fachverband Dynamik und Statistische Physik

DY 5: Poster Session III: Statistical Physics, Complex Fluids and Soft Matter

DY 5.9: Poster

Tuesday, September 28, 2021, 17:30–19:30, P

Diagrammatic expansion of the two-point effective action around non-Gaussian theories — •Tobias Kühn and Frédéric van Wijland — Laboratoire Matière et Systèmes Complexes, Université de Paris

Consider a many-body problem, such as one involving interacting spins, particles or the time series of a random signal, of which we know the corresponding one- and two-point correlation functions. We suppose that the distribution of the spin values (or particle positions,...) is written in a Boltzmann form with a Hamiltonian possessing one- and two-body interactions only. How does one choose the corresponding couplings so that the distribution generates the prescribed statistics? This so-called inverse problem is conveniently described by the second Legendre transform of the cumulant-generating function, the two-point effective action, whose arguments are means and correlations. The couplings we seek for are then explicitly given by its derivatives. Weak correlation approximations have been proven useful to compute the two-point effective action (Sessak & Monasson 2009). As long as the one-body problem is described by independent Gaussian distributions, they can be routinely derived using diagrammatic methods dating back to Feynman, Luttinger and Ward. Here we explain how similar diagrammatics can be extended to the case in which the one-body problem is not of a Gaussian nature. We discuss how this can prove useful in inference problems pertaining to neuroscience and other complex systems. Another possible application is the derivation of mean-field theories self-consistently taking into account pairwise correlations.